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Using concave envelopes to globally solve the nonlinear sum of ratios problem

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Abstract

This article presents a branch and bound algorithm for globally solving the nonlinear sum of ratios problem (P). The algorithm works by globally solving a sum of ratios problem that is equivalent to problem (P). In the algorithm, upper bounds are computed by maximizing concave envelopes of a sum of ratios function over intersections of the feasible region of the equivalent problem with rectangular sets. The rectangular sets are systematically subdivided as the branch and bound search proceeds. Two versions of the algorithm, with convergence results, are presented. Computational advantages of these algorithms are indicated, and some computational results are given that were obtained by globally solving some sample problems with one of these algorithms.

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References

  1. Almogy, Y. and Levin, O. (1970), Parametric analysis of a multi-stage stochastic shipping problem, Operational Research’ 69, Tavistok Publications, London.

    Google Scholar 

  2. Benson, H.P. (1996), Deterministic algorithms for constrained concave minimization: a unified critical survey, Naval Research Logistics 43: 765–795.

    Google Scholar 

  3. Benson, H.P. (2001), On the construction and utilization of convex and concave envelopes of bilinear and fractional functions, Working Paper, Department of Decision and Information Sciences, University of Florida, Gainesville, FL.

    Google Scholar 

  4. Benson, H.P. (1999), An outcome space branch and bound outer-approximation algorithm for convex multiplicative programming, Journal of Global Optimization 15: 315–342.

    Google Scholar 

  5. Benson, H.P. (1990), Separable concave minimization via partial outer approximation and branch and bound, Operations Reserch Letters 9: 389–394.

    Google Scholar 

  6. Cambini, A., Martein, L. and Schaible, S. (1989), On maximizing a sum of ratios, Journal of Information and Optimization Sciences 10: 65–79.

    Google Scholar 

  7. Colantoni, C.S., Manes, R.P. and Whinston, A. (1969), Programming, profit rates and pricing decisions, The Accounting Review 44: 467–481.

    Google Scholar 

  8. Dur, M., Horst, R. and Thoai, N.V. (2001), Solving sum-of-ratios fractional programs using efficient points, Optimization, to appear.

  9. Falk, J.E. and Palocsay, S.W. (1994), Image space analysis of generalized fractional programs, Journal of Global Optimization 4: 63–88.

    Google Scholar 

  10. Fourer, R., Gay, D.M. and Kernighan, B.W. (1993), AMPL, a Modeling Language for Mathematical Programming, Boyd and Fraser Publishing Company, Danvers, MA.

    Google Scholar 

  11. Freund, R. and Jarre, F. (2001), Solving the sum-of-ratios problem by an interior point method, Journal of Global Optimization 19: 83–102.

    Google Scholar 

  12. Gallo, G. and Ulkucu, A. (1977), Bilinear programming: an exact algorithm, Mathematical Programming 12: 173–194.

    Google Scholar 

  13. Horst, R. and Pardalos, P.M. (eds) (1995), Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands.

    Google Scholar 

  14. Horst, R. and Tuy, H. (1993), Global Optimization: Deterministic Approaches, Springer Verlag, Berlin.

    Google Scholar 

  15. Kalantari, B. and Rosen, J.B. (1987), An algorithm for global minimization of linearly constrained concave quadratic functions, Mathematics of Operations Research 12: 544–561.

    Google Scholar 

  16. Konno, H. and Abe, N. (1999), Minimization of the sum of three linear fractional functions, Journal of Global Optimization 15: 419–432.

    Google Scholar 

  17. Konno, H. and Fukaishi, K. (2000), A branch and bound algorithm for solving low rank linear multiplicative and fractional programming problems, Journal of Global Optimization 18: 283–299.

    Google Scholar 

  18. Konno, H. and Inori, M. (1989), Bond portfolio optimization by bilinear fractional programming, Journal of the Operations Research Society of Japan 32: 143–158.

    Google Scholar 

  19. Konno, H., Kuno, T. and Yajima, Y. (1994), Global minimization of a generalized convex multiplicative function, Journal of Global Optimization 4: 47–62.

    Google Scholar 

  20. Konno, H. and Watanabe, H. (1996), Bond portfolio optimization problems and their application to index tracking: a partial optimization approach, Journal of the Operations Research Society of Japan 39: 295–306.

    Google Scholar 

  21. Konno, H., Yajima, Y. and Matsui, T. (1991), Parametric simplex algorithms for solving a special class of nonconvex minimization problems, Journal of Global Optimization 1: 65–81.

    Google Scholar 

  22. Konno, H. and Yamashita, H. (1999), Minimizing sums and products of linear fractional functions over a polytope, Naval Research Logistics 46: 583–596.

    Google Scholar 

  23. Kuno, T. (2000), A branch-and-bound algorithm for maximizing the sum of several linear ratios, University of Tsukuba, Research Report ISE-TR-00–175, Tsukuba, Japan.

    Google Scholar 

  24. Murtagh, B.A. and Saunders, M.A. (1998), MINOS 5.5 User's Guide, Stanford University, Technical Report SOL 83–20R, Stanford, CA.

    Google Scholar 

  25. Quesada, I. and Grossman, I. (1995), A global optimization algorithm for linear fractional and bilinear programs, Journal of Global Optimization 6: 39–76.

    Google Scholar 

  26. Rosen, J.B. and Pardalos, P.M. (1986), Global minimization of large-scale constrained concave quadratic problems by separable programming, Mathematical Programming 34: 163–174.

    Google Scholar 

  27. Schaible, S. (1995), Fractional programming, in: R. Horst and P.M. Pardalos (eds), Handbook of Global Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands, 495–608.

    Google Scholar 

  28. Schaible, S. (1994), Fractional programming with sums of ratios, University of Pisa, Report No. 83, Pisa, Italy.

  29. Schaible, S. (1977), A note on the sum of a linear and linear-fractional function, Naval Research Logistics Quarterly 24: 691–693.

    Google Scholar 

  30. Tuy, H. (1998), Convex Analysis and Global Optimization, Kluwer Academic Publishers, Dordrecht, The Netherlands.

    Google Scholar 

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Authors and Affiliations

  1. University of Florida, Warrington College of Business Administration, Gainesville, FL, 32611, USA

    Harold P. Benson

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  1. Harold P. Benson
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Benson, H.P. Using concave envelopes to globally solve the nonlinear sum of ratios problem. Journal of Global Optimization 22, 343–364 (2002). https://doi.org/10.1023/A:1013869015288

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